Fourth Order Linear Recurrences Satisfied by Wythoff Pairs

A result of Stolarsky is extended to show that there are infinitely many irreducible fourth order linear recurrences satisfied by sequences of pairs $$(\left\lfloor {i\Phi } \right\rfloor ,\left\lfloor {i\Phi ^2 } \right\rfloor )$$ , where i is a natural number and φ is the golden ratio $$\frac{1}{2}(1 + \sqrt {5)} $$ . It is also proved that the characteristic polynomial of every such recurrence factorizes non-trivially if φ is adjoined to the rationals.